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Find the integral $\int\frac{x^2+3x}{4x^2+9}dx$

Step-by-step Solution

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Solution

Final Answer

$\frac{1}{4}x-\frac{3}{8}\arctan\left(\frac{2x}{3}\right)-\frac{3}{4}\ln\left(\frac{3}{\sqrt{4x^2+9}}\right)+C_0$
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Step-by-step Solution

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Divide $x^2+3x$ by $4x^2+9$

$\begin{array}{l}\phantom{\phantom{;}4x^{2}+9;}{\phantom{;}\frac{1}{4}\phantom{;}\phantom{;}}\\\phantom{;}4x^{2}+9\overline{\smash{)}\phantom{;}x^{2}+3x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}4x^{2}+9;}\underline{-x^{2}\phantom{-;x^n}-\frac{9}{4}\phantom{;}\phantom{;}}\\\phantom{-x^{2}-\frac{9}{4}\phantom{;}\phantom{;};}\phantom{;}3x\phantom{;}-\frac{9}{4}\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}4x^{2}+9;}{\phantom{;}\frac{1}{4}\phantom{;}\phantom{;}}\\\phantom{;}4x^{2}+9\overline{\smash{)}\phantom{;}x^{2}+3x\phantom{;}\phantom{-;x^n}}\\\phantom{\phantom{;}4x^{2}+9;}\underline{-x^{2}\phantom{-;x^n}-\frac{9}{4}\phantom{;}\phantom{;}}\\\phantom{-x^{2}-\frac{9}{4}\phantom{;}\phantom{;};}\phantom{;}3x\phantom{;}-\frac{9}{4}\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((x^2+3x)/(4x^2+9))dx. Divide x^2+3x by 4x^2+9. Resulting polynomial. Expand the integral \int\left(\frac{1}{4}+\frac{3x-\frac{9}{4}}{4x^2+9}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{4}dx results in: \frac{1}{4}x.

Final Answer

$\frac{1}{4}x-\frac{3}{8}\arctan\left(\frac{2x}{3}\right)-\frac{3}{4}\ln\left(\frac{3}{\sqrt{4x^2+9}}\right)+C_0$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve integral of ((x^2+3x)/(4x^2+9))dx using partial fractionsSolve integral of ((x^2+3x)/(4x^2+9))dx using basic integralsSolve integral of ((x^2+3x)/(4x^2+9))dx using u-substitutionSolve integral of ((x^2+3x)/(4x^2+9))dx using integration by partsSolve integral of ((x^2+3x)/(4x^2+9))dx using trigonometric substitution

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Function Plot

Plotting: $\frac{1}{4}x-\frac{3}{8}\arctan\left(\frac{2x}{3}\right)-\frac{3}{4}\ln\left(\frac{3}{\sqrt{4x^2+9}}\right)+C_0$

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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